[논문 리뷰] Constructal Evolution as a Nonsmooth Dynamical System: Stability and Selection of Flow Architectures

Pascal Stiefenhofer|arXiv (Cornell University)|2026. 03. 05.

Quantum chaos and dynamical systems인용 수 0

한 줄 요약

본 논문은 Constructal evolution을 Filippov nonsmooth dynamical system으로, compact한 건축적 상태 공간 위에서 제시하며, 소멸(dissipation)과 수축(contraction) 하에서 선택된 흐름 아키텍처의 존재성, 고유성, 전역 지수 안정성을 증명한다. Bejan의 area-to-point hierarchy를 동적 스위칭 및 슬라이딩 매니폴드로 포함시킨다.

ABSTRACT

Constructal Law states that a finite-size flow system that persists in time evolves its configuration so as to provide progressively easier access to the currents that flow through it. Classical Constructal theory derives hierarchical flow architectures from static resistance minimization under finite-size constraints, but many transport systems operate under irreversible limits that induce regime switching and discontinuous adjustment laws. We formulate Constructal evolution as an autonomous nonsmooth dynamical system. The architectural configuration is modeled as the state of a Filippov differential inclusion defined on a compact forward-invariant admissible set. Irreversible transport constraints generate switching manifolds across which the adjustment field is discontinuous. A resistance dissipation inequality encodes the Constructal principle of progressively improving access as a nonsmooth Lyapunov condition, while a uniform contraction assumption provides spectral bounds on the generalized Jacobians of the regime-dependent dynamics. Under these conditions we prove that the resulting inclusion admits a unique equilibrium architecture and that every admissible trajectory converges to it exponentially. Finite size, irreversibility, and resistance dissipation therefore imply existence, uniqueness, and global stability of persistent flow configurations without invoking static optimization. As an application, the classical area--to--point transport hierarchy of Bejan et. al. is embedded in the dynamical framework. The optimal assembly ratios appear as switching manifolds, while the classical scaling relations arise as sliding invariant sets of the Filippov inclusion. Their intersection defines the uniquely selected globally attracting architecture.

연구 동기 및 목표

• Motivate a dynamical formulation of Constructal Law that handles regime switching and finite-size constraints.
• Model architectural evolution as a Filippov differential inclusion on a compact, forward-invariant set.
• Introduce a resistance dissipation inequality that acts as a nonsmooth Lyapunov condition.
• Impose a uniform contraction condition to achieve architectural selection.
• Demonstrate dynamic embedding of the Bejan–Bădescu–De Vos area–to–point hierarchy as invariant switching/manifolds.

제안 방법

• Formulate architectural configurations as a state vector x(t) in a compact set K with a set-valued velocity F(x) (Filippov inclusion).
• Impose persistence via forward invariance and forward completeness using Nagumo viability.
• Define a resistance functional R on K and derive a nonsmooth dissipation inequality showing R decreases along trajectories.
• Use contraction theory in nonsmooth settings to obtain incremental exponential stability and uniqueness of the attracting architecture.
• Relate the dynamic equilibrium to sliding invariant sets and switching manifolds representing optimal ratios and classical scaling.

실험 결과

연구 질문

• RQ1 Can Constructal evolution be formulated as a globally well-posed nonsmooth dynamical system that accounts for irreversibility and regime switching?
• RQ2 Under what conditions do resistance dissipation and contraction guarantee existence, uniqueness, and global exponential convergence to a selected architecture?
• RQ3 How does the classical area–to–point transport hierarchy emerge dynamically as invariant structures within a Filippov framework?
• RQ4 What is the role of sliding motions and switching manifolds in enforcing dynamic architectural selection?

주요 결과

• A Filippov autonomous inclusion on a compact set yields existence, forward invariance, and forward completeness of architectural trajectories.
• A nonsmooth dissipation inequality ensures the resistance functional nonincreases along all trajectories, driving the system toward a stationary balance set.
• A uniform contraction condition yields incremental exponential stability, forcing all admissible trajectories to converge toward the same architecture.
• The classical Bejan–Bădescu–De Vos area-to-point hierarchy is embedded in the framework as switching manifolds and sliding invariant sets, whose intersection uniquely selects the globally attracting architecture.
• The combination of finite size, irreversibility, and resistance dissipation suffices to guarantee existence, uniqueness, and global stability of persistent flow configurations without static optimization.